In Fig. 8-52, a 3.5 kg block is accelerated from rest by a...

In Fig. 8-52, a $3.5 \mathrm{~kg}$ block is accelerated from rest by a compressed spring of spring constant $640 \mathrm{~N} / \mathrm{m}$. The block leaves the spring at the spring's relaxed length and then travels over a horizontal floor with a coefficient of kinetic friction $\mu_{k}=0.25 .$ The frictional force stops the block in distance $D=7.8 \mathrm{~m}$. What are (a) the increase in the thermal energy of the block-floor system, (b) the maximum kinetic energy of the block, and (c) the original compression distance of the spring?

Key Concepts

Conservation of Energy

This principle states that energy in a closed system is conserved and can change forms, such as transforming stored potential energy into kinetic energy or thermal energy. It is crucial for analyzing systems where energy is transferred or dissipated, ensuring that all energy inputs and outputs balance when accounting for losses like friction.

Spring Potential Energy

Spring potential energy is the energy stored within a spring when it is either compressed or stretched. Based on Hooke's law, this energy is a function of the spring constant and the displacement from its equilibrium position. It describes the work that can be done by the spring when it returns to its natural length.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion. It is determined by the object's mass and the square of its velocity. In scenarios where objects are accelerated, maximum kinetic energy represents the peak energy state before other forces, such as friction, begin to work on decelerating the object.

Work Done by Friction

Work done by friction refers to the process by which frictional forces convert the mechanical energy of a moving object into thermal energy. This concept is vital in problems where energy is dissipated as heat, such as when an object slides along a rough surface, reducing its kinetic energy.

Transcript

00:01 Well months off block is equal to 3.5 kilograms.

00:05 So massive block m is equal to 3.5 are kilograms on dhe supreme constant key is equal do ah, 640 newtons per meters on coefficient of kinetic friction, which is me.

00:28 Okay, is the coefficient of kinetic friction and it's equal to 0.25 and then we have the displacement d eyes equal to 7.8 meters from the release off the block to its rest.

00:52 Now let's soul part, eh? well, here we have ah ee.

01:00 Thermal is equal to the force of friction times the displacement and it's equal to well here.

01:15 F is the ah friction and d is the displacement and we need to the wanting more.

01:24 We also know that we're here.

01:26 A normal force are yeah, is equal to mm two.

01:38 It's equal to wait, and this is equal to a 3.5.

01:44 Multiply by 9.8 on this is equal to 34.3 newton's.

01:50 It's equal to 34 wind three nathan's on since force of friction f is equal to are times in uk, our time's meal okay and therefore, um, f is equal to, um 34 wind tree multiply by 0.25 and it's equal.

02:23 Do 0.58 newtons.

02:29 It 0.58 newton.

02:32 So the friction is 8.58 newtons.

02:36 Now we have here.

02:40 Eat thermal is equal to f times t know f is eight point 58 multiplied by d is equal to say 1.8.

02:50 And therefore so e caramel.

02:55 Ah, easy called 60.

02:59 It's equal to sixtus.

03:00 Expanded 9 to 66 0.89 jules.

03:06 All right, now let's sew part b...